Finite Group Databasebeta

← #28597147

Subgroups — #28597147

Order 15104.

1062 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837780C211yesmaximal, characteristic
2C2×#16837510C211yesmaximal, characteristic
3C2×#16837507C212yesmaximal
4C4×#14273725C212yesmaximal
5#21424968C211yesmaximal, characteristic
6#21424965C211yesmaximal, characteristic
7#21424964C212yesmaximal
8#21424896C212yesmaximal
9#21424251C211yesmaximal, characteristic
10#97×Dic59C211yesmaximal, characteristic
11C2×C59×#441C211yesmaximal, characteristic
12C22×Dic236C2211yescharacteristic
13C22×#11745672C2212yes—
14C2×#14273725C2212yes—
15C2×C4×Dic118C2212yes—
16#16837780C418yescomplemented
17#16837510—24no—
18#16837507—28no—
19C2×#14273659C2211yescharacteristic
20C2×#14273658C2212yes—
21C2×#14273652C2212yes—
22C2×#14273651C2212yes—
23C2×#14273650C2212yes—
24C2×#14273638C2211yescharacteristic
25C2×#14273637C2211yescharacteristic
26C2×#14273637C2211yescharacteristic
27C2×C8×Dic59C2211yescharacteristic
28C2×C8×Dic59C2211yescharacteristic
29#16837309C2212yes—
30#16837307C2212yes—
31#16837297—24no—
32#16837271C2212yes—
33C4×Dic59⋊C4—24no—
34C42×Dic59C2211yescharacteristic
35C22×Q16×C59C2211yescharacteristic
36C2×C4×Q8×C59C2212yes—
37C59×#441—24no—
38C2×#106×C59C2211yescharacteristic
39C2×#103×C59C2212yes—
40C2×#97×C59C2211yescharacteristic
41C22×Dic118C2312yes—
42C2×Dic236—22no—
43C2×Dic236—22no—
44C2×Dic236C2×C414yes—
45C2×#11745672—24no—
46C2×#11745672—24no—
47C2×#11745672C2×C418yes—
48C22×C59⋊C8C2311yescharacteristic
49C22×C59⋊C8C2311yescharacteristic
50#14273725—24no—
51#14273725C2×C418yes—
52C4×Dic118—24no—
53C4×Dic118—24no—
54C2×C4⋊Dic59C2312yes—
55C2×Dic59⋊C4—24no—
56C2×Dic59⋊C4—22no—
57C2×Dic59⋊C4—24no—
58C2×C4×Dic59—22no—
59C2×C4×Dic59D414yes—
60C2×C4×Dic59C2311yescharacteristic
61C2×C4×Dic59C2311yescharacteristic
62#14273659—22no—
63#14273658—24no—
64#14273652—24no—
65#14273651—24no—
66#14273650—24no—
67#14273638D414yes—
68#14273637—22no—
69#14273637—22no—
70C8×Dic59—22no—
71C8×Dic59C2×C414yes—
72C59⋊#95—24no—
73C59⋊#95—24no—
74C22×Q8×C59C2312yes—
75C2×Q16×C59—22no—
76C2×Q16×C59—22no—
77C2×Q16×C59C2×C414yes—
78C22×C8×C59C2311yescharacteristic
79C22×C8×C59C2311yescharacteristic
80C4×Q8×C59—24no—
81C4×Q8×C59—24no—
82C2×C4⋊C4×C59—22no—
83C2×C4⋊C4×C59C2312yes—
84C2×C42×C59—22no—
85C2×C42×C59C2311yescharacteristic
86#106×C59—22no—
87#103×C59—24no—
88#97×C59D414yes—
89C2×Q8×C59—22no—
90C2×Q8×C59—22no—
91C2×Q8×C59—22no—
92C2×Q8×C59—22no—
93C2×Q8×C59—22no—
94C2×Q8×C59C22×C414yes—
95C22×C4×C59—22no—
96C22×C4×C59—22no—
97C22×C4×C59C2×D412yes—
98C22×C4×C59C2411yescharacteristic
99C22×Dic59—22no—
100C22×Dic59—22no—
101C22×Dic59C2×D412yes—
102C22×Dic59C2×D412yes—
103C2×Dic118—22no—
104C2×Dic118—22no—
105C2×Dic118—22no—
106C2×Dic118—22no—
107C2×Dic118—22no—
108C2×Dic118C22×C414yes—
109Q16×C59—28no—
110C2×C8×C59—21no—
111C2×C8×C59—21no—
112C2×C8×C59C22×C412yes—
113C2×C8×C59C2×D412yes—
114C2×C8×C59C2×D412yes—
115C2×C8×C59#4112yes—
116C4⋊C4×C59—22no—
117C4⋊C4×C59—22no—
118C4⋊C4×C59—44no—
119C42×C59—44no—
120C42×C59C2×D412yes—
121C42×C59C2×D412yes—
122#11745672—216no—
123C4⋊Dic59—22no—
124C4⋊Dic59—22no—
125Dic59⋊C4—216no—
126Dic59⋊C4—44no—
127Dic59⋊C4—48no—
128C4×Dic59—21no—
129C4×Dic59—21no—
130C4×Dic59—28no—
131C4×Dic59—44no—
132C4×Dic59C22×C412yes—
133C4×Dic59#4112yes—
134C2×C59⋊C8—21no—
135C2×C59⋊C8—21no—
136C2×C59⋊C8C22×C412yes—
137C2×C59⋊C8C2×D412yes—
138C2×C59⋊C8C2×D412yes—
139C2×C59⋊C8#4112yes—
140Dic236—28no—
141C23×C59C22×D411yescharacteristic
142Q8×C59—24no—
143Q8×C59—24no—
144Q8×C59—42no—
145Q8×C59—42no—
146C2×C4×C59—24no—
147C2×C4×C59—24no—
148C2×C4×C59—42no—
149C2×C4×C59—42no—
150C2×C4×C59—42no—
151C2×C4×C59—44no—
152C2×C4×C59#12714yes—
153C2×C4×C59C23×C411yescharacteristic
154C2×C4×C59C22×D411yescharacteristic
155C2×C4×C59C22×D411yescharacteristic
156C2×C4×C59C2×#4111yescharacteristic
157C2×C4×C59C2×#4111yescharacteristic
158C2×C4×C59C2×#4111yescharacteristic
159C2×Dic59—24no—
160C2×Dic59—22no—
161C2×Dic59—24no—
162C2×Dic59—42no—
163C2×Dic59—42no—
164C2×Dic59—42no—
165C2×Dic59—44no—
166C2×Dic59C4×D414yes—
167C2×Dic59#13718yes—
168Dic118—24no—
169Dic118—24no—
170Dic118—42no—
171Dic118—42no—
172C8×C59—22no—
173C8×C59C4×D414yes—
174C59⋊C8—22no—
175C59⋊C8C4×D414yes—
176#21691—591nomaximal, Sylow
177C22×C59C2×C4×D411yescharacteristic
178C22×C59C2×#12711yescharacteristic
179C22×C59#53211yescharacteristic
180C22×C59C2×#13714yes—
181C4×C59—24no—
182C4×C59—44no—
183C4×C59—82no—
184C4×C59C2×C4×D411yescharacteristic
185C4×C59C2×C4×D411yescharacteristic
186C4×C59#53211yescharacteristic
187C4×C59#53211yescharacteristic
188Dic59—44no—
189Dic59—42no—
190Dic59—82no—
191Dic59#44118yescomplemented
192C2×#494—591no—
193C2×#441—591no—
194C2×#441—592no—
195C2×#441D5911yescharacteristic
196C4×C4⋊Q8—592no—
197#1971—591no—
198#1971—592no—
199#1971—591no—
200#1956—592no—
201#1844—591no—
202C4×#97—591no—
203C2×C59C4×#12711yescharacteristic
204C2×C59C2×#44112yes—
205C2×C59#315212yes—
206C2×C59#318512yes—
207C22×Q16—592no—
208C22×Q16—591no—
209C22×Q16C2×D5911yescharacteristic
210C2×C4⋊Q8—592no—
211C2×C4×Q8—592no—
212C2×C4×Q8C2×D5912yes—
213#494—598no—
214#441—24no—
215#441—1188no—
216#441—1184no—
217C2×#106—591no—
218C2×#106—592no—
219C2×#106C2×D5911yescharacteristic
220C2×#103—592no—
221C2×#103—592no—
222C2×#103—592no—
223C2×#103C2×D5912yes—
224C2×#97—591no—
225C2×#97—591no—
226C2×#97—591no—
227C2×#97C2×D5911yescharacteristic
228C2×C4×C8—591no—
229C2×C4×C8—591no—
230#391—592no—
231#391—592no—
232#389—1184no—
233#384—592no—
234C4×C4⋊C4—1184no—
235C43—591no—
236C59#2169111yescharacteristic, complemented, Sylow
237C22×Q8—592no—
238C22×Q8C22×D5912yes—
239C2×Q16—22no—
240C2×Q16—22no—
241C2×Q16—598no—
242C2×Q16—594no—
243C2×Q16—1184no—
244C2×Q16—1184no—
245C2×Q16—1182no—
246C2×Q16—1182no—
247C2×Q16C4×D5914yes—
248C22×C8—591no—
249C22×C8—591no—
250C22×C8C22×D5911yescharacteristic
251C22×C8C22×D5911yescharacteristic
252C4⋊Q8—598no—
253C4⋊Q8—1184no—
254C4×Q8—24no—
255C4×Q8—24no—
256C4×Q8—1184no—
257C4×Q8—1184no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—592no—
260C2×C4⋊C4—1184no—
261C2×C4⋊C4—1184no—
262C2×C4⋊C4—1182no—
263C2×C4⋊C4C22×D5912yes—
264C2×C42—22no—
265C2×C42—591no—
266C2×C42—594no—
267C2×C42—591no—
268C2×C42—1182no—
269C2×C42C22×D5911yescharacteristic
270#106—22no—
271#106—1184no—
272#106—1182no—
273#103—24no—
274#103—1184no—
275#103—1184no—
276#103—1184no—
277#97—594no—
278#97—1182no—
279#97—1182no—
280#97C22⋊D5914yes—
281C4×C8—594no—
282C4×C8—1182no—
283#95—1184no—
284#95—1184no—
285C2×Q8—22no—
286C2×Q8—22no—
287C2×Q8—22no—
288C2×Q8—22no—
289C2×Q8—22no—
290C2×Q8—594no—
291C2×Q8—1182no—
292C2×Q8—1182no—
293C2×Q8—1182no—
294C2×Q8—1182no—
295C2×Q8—1182no—
296C2×Q8C2×C4×D5914yes—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—592no—
300C22×C4—592no—
301C22×C4—1182no—
302C22×C4—1182no—
303C22×C4D4×D5912yes—
304C22×C4C23×D5911yescharacteristic
305Q16—28no—
306Q16—11816no—
307Q16—1188no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—592no—
311C2×C8—592no—
312C2×C8—592no—
313C2×C8—592no—
314C2×C8—1181no—
315C2×C8—1181no—
316C2×C8C2×C4×D5912yes—
317C2×C8#1174568512yes—
318C2×C8C2×C22⋊D5912yes—
319C2×C8C2×C22⋊D5912yes—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—44no—
323C4⋊C4—11816no—
324C4⋊C4—1182no—
325C4⋊C4—1182no—
326C4⋊C4—2368no—
327C4⋊C4—2364no—
328C42—44no—
329C42—592no—
330C42—592no—
331C42—1188no—
332C42—1181no—
333C42—1181no—
334C42—2364no—
335C42D4×D5912yes—
336C42C2×C22⋊D5912yes—
337C23C2×D4×D5911yescharacteristic
338Q8—24no—
339Q8—24no—
340Q8—42no—
341Q8—42no—
342Q8—1184no—
343Q8—1184no—
344Q8—2362no—
345Q8—2362no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—44no—
351C2×C4—42no—
352C2×C4—598no—
353C2×C4—594no—
354C2×C4—1184no—
355C2×C4—1182no—
356C2×C4—1184no—
357C2×C4—2364no—
358C2×C4—2362no—
359C2×C4—2362no—
360C2×C4—2362no—
361C2×C4#1427372314yes—
362C2×C4C22×C4×D5911yescharacteristic
363C2×C4C2×D4×D5911yescharacteristic
364C2×C4C22×C22⋊D5911yescharacteristic
365C2×C4#41×D5911yescharacteristic
366C2×C4#41×D5911yescharacteristic
367C2×C4C2×#1174568511yescharacteristic
368C8—22no—
369C8—594no—
370C8—1182no—
371C8C4×C22⋊D5914yes—
372C22C4×D4×D5911yescharacteristic
373C22C2×#1427372311yescharacteristic
374C22#1683794911yescharacteristic
375C22#137×D5914yes—
376C4—24no—
377C4—44no—
378C4—82no—
379C4—598no—
380C4—4722no—
381C4—2362no—
382C4—2364no—
383C4C4×D4×D5911yescharacteristic
384C4C2×C4×C22⋊D5911yescharacteristic
385C4#1683794811yescharacteristic
386C4#1683794911yescharacteristic
387C2C4×#1427372311yescharacteristic
388C2#441×D5912yes—
389C2#2143455112yes—
390C2#2143483812yes—
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC2×C59×#441C2
socle, prime radicalC23×C59C22×D4
cosocleC22×C4×C59C24
hypercenterC2×#441D59
hypocenterC59#21691
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#441

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
16. #16837780C4semidirect
191. Dic59#441semidirect
236. C59#21691semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168377802
C2×#168375102
C2×#168375072
C4×#142737252
#214249682
#214249652
#214249642
#214248962
#214242512
#97×Dic592
C2×C59×#4412
#2169159